I want to find two connected, non-isomorphic graphs with order n with the same eccentricity sequence where $n\ge 4$.
The closest I could think of was $K_{1,n}$ having an eccentricity sequence of ${1,2,2,2,2,2…,2}$ and $K_{n+1}$ having an eccentricity sequence of ${1,1,1,1,1,1,1….,1}$. This yields eccentricity sequence being off by $1$ for $n\ge 4$.
Is there some methodical way of finding two non-isomorphic graphs with the same eccentricity sequence?
Best Answer
Try $K_{1,n}$ and $K_{1,n}+e$ (add one more edge to $K_{1,n}$) where $n\ge3$.